Gromov-Hausdorff distances from simply connected geodesic spaces to the circle.

In the following, for $n\geq 1$ we give the unit sphere $\mathbb{S}^{n}:=\{z\in\mathbb{R}^{n+1};\|z\|=1\}$ its intrinsic metric $d_{\mathbb{S}^{n}}$. That is, for any two points $p,q\in\mathbb{S}^{n}$ seen as vectors in $\mathbb{R}^{n+1}$, $d_{\mathbb{S}^{n}}(p,q)$ is the angle between $p$ and $q$.

The Gromov-Hausdorff (GH) distance provides a quantitative measure of how far two metric spaces are from being isometric. Since being introduced by Edwards ([Ed], 1975) and Gromov ([Gr1, Gr2], 1981), it has proved useful in the study of, for example, shapes formed by point cloud data [MS, BBK], convergence results for sequences of Riemannian manifolds [CC, PW, C1, C2], differentiability results in metric measure spaces [Ke, Ch] or the stability of topological invariants of metric spaces under small deformations (see [Pe] and the present article).

However, it is hard to find the exact value of the GH distance between two given metric spaces. There have been recent efforts to determine the precise value of the GH distance between spheres. In [LMS, Theorem B] it is proved that $d_{\textup{GH}}(\mathbb{S}^{n},\mathbb{S}^{m})\geq\frac{1}{2}\arccos\left(\frac{-1}{m+1}\right)$ for all $n>m$ which implies that

for all $n\geq 2$. Via these lower bounds and matching upper bounds, the authors find the precise value of the GH distance for the pairs $(\mathbb{S}^{1},\mathbb{S}^{2})$, $(\mathbb{S}^{1},\mathbb{S}^{3})$, and $(\mathbb{S}^{2},\mathbb{S}^{3})$. Similarly, in [Ka, Lemma 2.3] it was proved that for any interval $I\subseteq\mathbb{R}$ we have $d_{\textup{GH}}(I,\mathbb{S}^{1})\geq\frac{\pi}{3}$ (see also [LMS, Prop. B.1]; in [JT, Theorem 6.8] the exact GH distances from $\mathbb{S}^{1}$ to intervals of any length $\lambda\in[0,\infty)$ are computed). In [ABC] the authors identify lower bounds for $d_{\textup{GH}}(\mathbb{S}^{m},\mathbb{S}^{n})$ that are often tighter than the ones from [LMS].

Via considerations related to persistent homology and the filling radius, in [LMO, Remark 9.19] it was deduced that for any compact geodesic metric space $X$ one has $d_{\textup{GH}}(X,\mathbb{S}^{1})\geq\frac{\pi}{6}$ and, as a generalization of Equation (1), the following conjecture was formulated:

In 1.1, the condition that $X$ is geodesic is necessary to relate the metric of $X$ to the fact that it is simply connected; if not, we could consider the space $X=\mathbb{S}^{1}\setminus\{1\}$, with its metric inherited from $\mathbb{S}^{1}$. Then $X$ is simply connected, but we clearly have $d_{\textup{GH}}(X,\mathbb{S}^{1})=0$. The present article gives a complete answer to 1.1; we state the main results after recalling some standard notation.

Let $(X,d_{X})$ be a metric space, and let $A,B\subseteq X$ be nonempty and let $x\in X$. The distance from $x$ to $A$ is defined as $d_{X}(x,A)=\inf_{a\in A}d_{X}(x,a)$, and the Hausdorff distance between $A$ and $B$ is given by

Given two nonempty metric spaces $(X,d_{X})$ and $(Y,d_{Y})$, we write $(X,d_{X})\cong(Y,d_{Y})$, or just $X\cong Y$ when the metrics are clear, whenever $(X,d_{X})$ and $(Y,d_{Y})$ are isometric. The Gromov-Hausdorff (GH) distance between $X$ and $Y$ is defined as the value in $[0,\infty]$ given by

Recall that a metric space $(X,d_{X})$ is a length space when for all $x,x^{\prime}\in X$, the distance $d_{X}(x,x^{\prime})$ is the infimum of lengths of paths $\gamma:[0,1]\to X$ with $\gamma(0)=x$ and $\gamma(1)=x^{\prime}$ (cf. [BBI, Chapter 2]). If in addition we have that for all $x,x^{\prime}\in X$ there is a path $\gamma$ from $x$ to $x^{\prime}$ with length $d_{X}(x,x^{\prime})$, we say that $X$ is geodesic.

To prove 1.4 from Theorem 1.3 note that every loop $\alpha:[0,1]\to\mathbb{S}^{1}$ with diameter $<\pi$ is nulhomotopic. So if there was a simply connected space $X$ with $d_{\textup{GH}}(X,\mathbb{S}^{1})<\frac{\pi}{4}$, then we could choose the constant $D$ from Theorem 1.3 to be just below $\frac{\pi}{4}$, which would give a contradiction as $\pi_{1}(\mathbb{S}^{1})\neq 0$.

Section 2 is almost entirely devoted to the proof of Theorem 1.3. At the end of the section we also give a lower bound for the distances from simply connected length spaces to $\mathbb{S}^{1}$ with the metric inherited from $\mathbb{R}^{2}$.

Section 3 is devoted to the construction of a simply connected geodesic space E with $d_{\textup{GH}}(\mathrm{E},\mathbb{S}^{1})\leq\frac{\pi}{4}$. This, in conjunction with 1.4, proves Theorem 1.2. In Section 3 we will not use the definition of Gromov-Hausdorff distance given above; we now recall (cf. §7.3 in [BBI]) an equivalent definition based on correspondences between sets, which will be more convenient for our purposes.

Given two sets $X$ and $Y$, we say a relation $R\subseteq X\times Y$ is a correspondence between $X$ and $Y$ if $\pi_{X}(R)=X$ and $\pi_{Y}(R)=Y$, where $\pi_{X}:X\times Y\to X$ and $\pi_{Y}:X\times Y\to Y$ are the coordinate projections. If $(X,d_{X})$ and $(Y,d_{Y})$ are metric spaces, we define the distortion of a nonempty relation $R\subseteq X\times Y$ as

In [BBI, Theorem 7.3.25] it is proved that, if $(X,d_{X}),(Y,d_{Y})$ are nonempty metric spaces, then

In Section 4 we adress the question of whether the space $\mathrm{E}$ we construct to prove Theorem 1.2, which is a compact $\mathbb{R}$-tree, is the unique space satisfying Theorem 1.2. It turns out that it is not hard to use E to construct a big family of spaces that satisfy Theorem 1.2 (see 4.2), and that there are $\mathbb{R}$-trees which satisfy Theorem 1.2 and are not isometric to E (see Note 4.6). However, in 4.5 we prove a minimality result for our space E: it is, up to isometry, the only complete $\mathbb{R}$-tree of length at most $\frac{5\pi}{4}$ which satisfies Theorem 1.2.

Acknowledgements. Special thanks to Facundo Mémoli for introducing the author to the topic of Gromov-Hausdorff distances and providing guidance while writing this article. The author gratefully acknowledges support from the grants BSF 2020124 and NSF CCF AF 2310412.

Figure 2 shows two metric graphs (see [BBI, Definition 3.2.9.] for a definition of metric graph) which we will call E and $\mathbb{S}^{1}$ for obvious reasons, with all edges having length $\frac{\pi}{4}$. The graph E has five edges, $a,b,c,d$ and $e$. We have also labelled the eight edges of $\mathbb{S}^{1}$ with the letters $a,b,c,d,e$.

We define a function $\Phi:\mathbb{S}^{1}\to\textup{E}$ which sends each of the $8$ edges of $\mathbb{S}^{1}$ isometrically to one of the $5$ segments $a,b,c,d,e$ of E as indicated by the labels and orientations in the figure. This specification of $\Phi$ gives two possible values $P,Q\in\textup{E}$ to $\Phi(1)$; for definiteness we choose $\Phi(1)=P$. Note that $\Phi$ is locally $1$-Lipschitz except at $1$, where there is a jump of length $\frac{\pi}{2}$ from $Q$ to $P$.

Note that, by Equation 3, the claim above implies that $d_{\textup{GH}}(\textup{E},\mathbb{S}^{1})\leq\frac{\pi}{4}$. Since E is a simply connected geodesic space, by 1.4 we obtain $d_{\textup{GH}}(\textup{E},\mathbb{S}^{1})=\frac{\pi}{4}$, which is the goal of this section.

It is natural to ask whether $\mathrm{E}$ is, up to isometry, the only geodesic simply connected space with GH distance $\frac{\pi}{4}$ to $\mathbb{S}^{1}$. We now use $\mathrm{E}$ to obtain a big family of simply connected geodesic spaces which are exactly at GH distance $\frac{\pi}{4}$ from $\mathbb{S}^{1}$.

So E is not the only geodesic, simply connected space at distance $\frac{\pi}{4}$ from $\mathbb{S}^{1}$. However, we prove below that E is the only complete tree with minimal length at distance $\frac{\pi}{4}$ from $\mathbb{S}^{1}$; to formalize this statement we will need to introduce some concepts and notation:

Note that all $\mathbb{R}$-trees are geodesic by definition, and the distance between any two points $x,x^{\prime}$ of an $\mathbb{R}$-tree is the length of the arc $[x,x^{\prime}]$. Moreover, any $\mathbb{R}$-tree $X$ is contractible: indeed, for any $x_{0}\in X$ we can define a homotopy $(f_{t})_{t\in[0,1]}$, $f_{t}:X\to X$, from a constant map to Id_X by letting $f_{t}(x)$ be the point in $[x,x_{0}]$ at distance $t\cdot d_{X}(x,x_{0})$ from $x_{0}$. Thus, by 1.4, any $\mathbb{R}$-tree $X$ satisfies $d_{\mathrm{GH}}(X,\mathbb{S}^{1})\geq\frac{\pi}{4}$.

Thus, 4.5 may be informally expressed as ‘besides $\mathrm{E}$, no other complete $\mathbb{R}$-tree $X$ of length $\leq\frac{5\pi}{4}$ can satisfy conditions 1 and 2 simultaneously’. If we remove condition 2, it is clear that many complete $\mathbb{R}$-trees of length $\leq\frac{5\pi}{4}$ can satisfy condition 1 (e.g. the $\mathbb{R}$-tree formed by a single point). So we certainly need to use condition 2 in order to prove our theorem. However, one may at first think that condition 1 is not necessary to prove our uniqueness theorem, that is, that any $\mathbb{R}$-tree $X$ of length $\leq\frac{5\pi}{4}$ satisfying condition 2 is isometric to $\mathrm{E}$. That is, ‘if an $\mathbb{R}$-tree $X$ is smaller than $E$, then we cannot place $X$ and $\mathbb{S}^{1}$ in such a way that all points of $\mathbb{S}^{1}$ are at distance $\leq\frac{\pi}{4}$ of $X$’. This is, however, not true. Indeed, consider the $\mathbb{R}$-tree $X$ shown below, which has four edges of length $\frac{\pi}{4}$ and four extreme points $p_{1},p_{2},p_{3},p_{4}$. Then $X$ has length $\pi$, but if we divide $\mathbb{S}^{1}$ into four intervals $I_{1},I_{2},I_{3},I_{4}$ of length $\frac{\pi}{2}$, one can give the disjoint union $\mathbb{S}^{1}\sqcup X$ a metric in which, for each $i=1,2,3,4$, all the points of $I_{i}$ are at distance $\frac{\pi}{4}$ from $p_{i}$.

The proof of 4.5 comprises the rest of this section. Fix a complete $\mathbb{R}$-tree $(X,d_{X})$ of length at most $\frac{5\pi}{4}$ satisfying $d_{\mathrm{GH}}(X,\mathbb{S}^{1})=\frac{\pi}{4}$.